Space of modular forms of level 336, weight 6, and Character 336.q

• Label: 336.6.q
• Level: $336$
• Weight: $6$
• Character orbit: 336.q
• Rep. character: $\chi_{336}(193,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $80$
• Newform subspaces: $14$
• Sturm bound: $384$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.q (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(384\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(336, [\chi])\).

	Total	New	Old
Modular forms	664	80	584
Cusp forms	616	80	536
Eisenstein series	48	0	48

Trace form

\( 80 q - 18 q^{3} + 62 q^{7} - 3240 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(336, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
336.6.q.a	$336$	$6$	336.q	7.c	$3$	$2$	$1$	$53.889$	\(\Q(\sqrt{-3}) \)	None					42.6.e.b	$2$	$0$	\(0\)	\(-9\)	\(-86\)	\(-49\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\zeta_{6})q^{3}-86\zeta_{6}q^{5}+(-98+\cdots)q^{7}+\cdots\)
336.6.q.b	$336$	$6$	336.q	7.c	$3$	$2$	$1$	$53.889$	\(\Q(\sqrt{-3}) \)	None					21.6.e.a	$2$	$0$	\(0\)	\(-9\)	\(-11\)	\(-259\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\zeta_{6})q^{3}-11\zeta_{6}q^{5}+(-133+\cdots)q^{7}+\cdots\)
336.6.q.c	$336$	$6$	336.q	7.c	$3$	$2$	$1$	$53.889$	\(\Q(\sqrt{-3}) \)	None					42.6.e.a	$2$	$0$	\(0\)	\(9\)	\(6\)	\(-119\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\zeta_{6})q^{3}+6\zeta_{6}q^{5}+(-126+133\zeta_{6})q^{7}+\cdots\)
336.6.q.d	$336$	$6$	336.q	7.c	$3$	$2$	$1$	$53.889$	\(\Q(\sqrt{-3}) \)	None					84.6.i.a	$2$	$0$	\(0\)	\(9\)	\(69\)	\(-245\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\zeta_{6})q^{3}+69\zeta_{6}q^{5}+(-147+\cdots)q^{7}+\cdots\)
336.6.q.e	$336$	$6$	336.q	7.c	$3$	$4$	$2$	$53.889$	\(\Q(\sqrt{-3}, \sqrt{-83})\)	None					21.6.e.b	$2$	$0$	\(0\)	\(-18\)	\(33\)	\(350\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+9\beta _{1}q^{3}+(20+20\beta _{1}+7\beta _{3})q^{5}+\cdots\)
336.6.q.f	$336$	$6$	336.q	7.c	$3$	$4$	$2$	$53.889$	\(\Q(\sqrt{-3}, \sqrt{9601})\)	None					42.6.e.c	$2$	$0$	\(0\)	\(-18\)	\(53\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\beta _{2})q^{3}+(\beta _{1}+26\beta _{2})q^{5}+\cdots\)
336.6.q.g	$336$	$6$	336.q	7.c	$3$	$4$	$2$	$53.889$	\(\Q(\sqrt{-3}, \sqrt{7081})\)	None					84.6.i.b	$2$	$0$	\(0\)	\(18\)	\(-47\)	\(174\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\beta _{2})q^{3}+(\beta _{1}-24\beta _{2})q^{5}+(73+\cdots)q^{7}+\cdots\)
336.6.q.h	$336$	$6$	336.q	7.c	$3$	$4$	$2$	$53.889$	\(\Q(\sqrt{-3}, \sqrt{505})\)	None					42.6.e.d	$2$	$0$	\(0\)	\(18\)	\(-17\)	\(408\)	$7$	$\mathrm{SU}(2)[C_{3}]$	\(q+9\beta _{1}q^{3}+(-8+7\beta _{1}+2\beta _{2}-3\beta _{3})q^{5}+\cdots\)
336.6.q.i	$336$	$6$	336.q	7.c	$3$	$8$	$4$	$53.889$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					84.6.i.c	$2$	$0$	\(0\)	\(-36\)	\(0\)	\(42\)	$2^{6}\cdot 3^{3}\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	\(q+9\beta _{1}q^{3}+(-\beta _{2}-\beta _{3})q^{5}+(-10+\cdots)q^{7}+\cdots\)
336.6.q.j	$336$	$6$	336.q	7.c	$3$	$8$	$4$	$53.889$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					21.6.e.c	$2$	$0$	\(0\)	\(36\)	\(0\)	\(-258\)	$2^{8}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\beta _{1})q^{3}+(\beta _{4}-\beta _{6})q^{5}+(-9+\cdots)q^{7}+\cdots\)
336.6.q.k	$336$	$6$	336.q	7.c	$3$	$8$	$4$	$53.889$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					168.6.q.a	$2$	$0$	\(0\)	\(36\)	\(64\)	\(-42\)	$2^{6}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9-9\beta _{1})q^{3}+(2^{4}\beta _{1}+\beta _{2})q^{5}+(-22+\cdots)q^{7}+\cdots\)
336.6.q.l	$336$	$6$	336.q	7.c	$3$	$10$	$5$	$53.889$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None					168.6.q.c	$2$	$0$	\(0\)	\(-45\)	\(-6\)	\(97\)	$2^{12}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-9\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{6})q^{5}+(14+\cdots)q^{7}+\cdots\)
336.6.q.m	$336$	$6$	336.q	7.c	$3$	$10$	$5$	$53.889$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					168.6.q.b	$2$	$0$	\(0\)	\(45\)	\(-75\)	\(113\)	$2^{12}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(9+9\beta _{1})q^{3}+(15\beta _{1}+\beta _{2}-\beta _{4})q^{5}+\cdots\)
336.6.q.n	$336$	$6$	336.q	7.c	$3$	$12$	$6$	$53.889$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None			✓		168.6.q.d	$2$	$0$	\(0\)	\(-54\)	\(17\)	\(-144\)	$2^{18}\cdot 3\cdot 7^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\beta _{1})q^{3}+(3\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(336, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)